cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A319161 Numbers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

Numbers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

Examples

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (211), (6), (1111), (7), (221), (8), (311), (9), (2111), (33), (222), (411).

Crossrefs

Cf. A001597, A007916, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A305732, A319151, A319160, A319163, A319165.

Programs

  • Mathematica
    Select[Range[100],GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1&]

A319162 Number of periodic integer partitions of n whose multiplicities are aperiodic, meaning the multiplicities of these multiplicities are relatively prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 6, 1, 9, 1, 12, 6, 16, 1, 27, 1, 33, 12, 46, 1, 70, 5, 84, 22, 110, 1, 172, 1, 188, 46, 251, 15, 366, 1, 418, 84, 540, 1, 775, 1, 863, 162, 1095, 1, 1535, 11, 1750, 251, 2154, 1, 2963, 49, 3323, 418, 4106, 1, 5567
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Examples

			The a(12) = 9 partitions:
  (66),
  (444), (441111),
  (3333), (33111111),
  (222222), (222111111), (2211111111),
  (111111111111).

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319163, A319164.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[GCD@@Sort[Length/@Split[#]]>1,GCD@@Length/@Split[Sort[Length/@Split[#]]]==1]&]],{n,30}]

A319164 Number of integer partitions of n that are neither relatively prime nor aperiodic.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 5, 0, 2, 2, 5, 0, 6, 0, 9, 2, 2, 0, 17, 1, 2, 3, 17, 0, 18, 0, 22, 2, 2, 2, 48, 0, 2, 2, 48, 0, 34, 0, 58, 11, 2, 0, 111, 1, 14, 2, 103, 0, 65, 2, 141, 2, 2, 0, 264, 0, 2, 19, 231, 2, 116, 0, 299, 2, 42
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

A partition is aperiodic if its multiplicities are relatively prime.

Examples

			The a(24) = 17 integer partitions:
  (12,12),
  (8,8,8),
  (6,6,6,6), (8,8,4,4), (9,9,3,3), (10,10,2,2),
  (4,4,4,4,4,4), (6,6,3,3,3,3), (6,6,4,4,2,2), (6,6,6,2,2,2), (8,8,2,2,2,2),
  (3,3,3,3,3,3,3,3), (4,4,4,4,2,2,2,2), (6,6,2,2,2,2,2,2),
  (4,4,4,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2,2,2).

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319165.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[GCD@@#>1,GCD@@Length/@Split[#]>1]&]],{n,30}]

A319810 Number of fully periodic integer partitions of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 8, 7, 11, 2, 17, 2, 18, 9, 15, 2, 32, 5, 22, 12, 34, 2, 54, 2, 49, 16, 51, 10, 94, 2, 77, 23, 112, 2, 152, 2, 148, 47, 165, 2, 258, 7, 247, 52, 286, 2, 400, 17, 402, 78, 439, 2, 657, 2, 594, 131, 711, 24
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2018

Keywords

Comments

An integer partition is fully periodic iff either it is a singleton or it is a periodic partition (meaning its multiplicities have a common divisor > 1) with fully periodic multiplicities.

Examples

			The a(12) = 11 fully periodic integer partitions:
  (12)
  (6,6)
  (4,4,4)
  (5,5,1,1)
  (4,4,2,2)
  (3,3,3,3)
  (3,3,3,1,1,1)
  (3,3,2,2,1,1)
  (2,2,2,2,2,2)
  (2,2,2,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)
Periodic partitions missing from this list are:
  (4,4,1,1,1,1)
  (3,3,1,1,1,1,1,1)
  (2,2,2,1,1,1,1,1,1)
  (2,2,1,1,1,1,1,1,1,1)
The first non-uniform fully periodic partition is (4,4,3,3,2,2,2,2,1,1,1,1).
The first periodic integer partition that is not fully periodic is (2,2,1,1,1,1).

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319811.

Programs

  • Mathematica
    totperQ[m_]:=Or[Length[m]==1,And[GCD@@Length/@Split[Sort[m]]>1,totperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],totperQ]],{n,30}]

A319179 Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 6, 1, 9, 1, 14, 7, 17, 1, 32, 1, 36, 15, 55, 1, 77, 6, 100, 27, 121, 1, 200, 1, 209, 56, 296, 19, 403, 1, 489, 101, 596, 1, 885, 1, 947, 192, 1254, 1, 1673, 14, 1979, 297, 2336, 1, 3300, 60, 3594, 490, 4564, 1, 5988, 1, 6841, 800
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2018

Keywords

Comments

An integer partition is aperiodic if its multiplicities are relatively prime.

Examples

			The a(12) = 9 integer partitions that are relatively prime but not aperiodic:
  (5511),
  (332211), (333111), (441111),
  (22221111), (33111111),
  (222111111),
  (2211111111),
  (111111111111).
The a(12) = 9 integer partitions that are aperiodic but not relatively prime:
  (12),
  (8,4), (9,3), (10,2),
  (6,3,3), (6,4,2), (8,2,2),
  (6,2,2,2),
  (4,2,2,2,2).

Crossrefs

Cf. A000837, A018783, A047966, A100953, A305563, A319149, A319160, A319162, A319164.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]>1]&]],{n,30}]

A319811 Number of totally aperiodic integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 99, 117, 162, 203, 286, 333, 469, 558, 737, 903, 1196, 1414, 1860, 2232, 2839, 3422, 4359, 5144, 6531, 7762, 9617, 11479, 14182, 16715, 20630, 24333, 29569, 34890, 42335, 49515, 59871, 70042, 83810, 98105, 117152
Offset: 1

Views

Author

Gus Wiseman, Sep 28 2018

Keywords

Comments

An integer partition is totally aperiodic iff either it is strict or it is aperiodic with totally aperiodic multiplicities.

Examples

			The a(6) = 7 aperiodic integer partitions are: (6), (51), (42), (411), (321), (3111), (21111). The first aperiodic integer partition that is not totally aperiodic is (432211).

Crossrefs

Cf. A000837, A018783, A047966, A098859, A100953, A305563, A319149, A319160, A319162, A319163, A319164, A319810.

Programs

  • Mathematica
    totaperQ[m_]:=Or[UnsameQ@@m,And[GCD@@Length/@Split[Sort[m]]==1,totaperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],totaperQ]],{n,30}]
Showing 1-6 of 6 results.

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